Mathematical and Statistical Methods for Actuarial Sciences and Finance

A Monte Carlo approach to value exchange options using a single stochastic factor 307

• N(d)is the cumulative standard normal distribution.

The typical simulation approach is to price the SEEO as the expectation value of
discounted cash-flows under the risk-neutral probabilityQ. So, for the risk-neutral
version of Equations (1) and (2), it is enough to replace the expected rates of returnμi
by the risk-free interest raterplus the premium-risk, namelyμi=r+λiσi,where
λiis the asset’s market price of risk, fori=V,D.So, we obtain the risk-neutral
stochastic equations:

dV

V

=(r−δv)dt+σv(dZv+λvdt)=(r−δv)dt+σvdZv∗, (5)

dD

D

=(r−δd)dt+σd(dZd+λddt)=(r−δd)dt+σddZd∗. (6)

The Brownian processesdZv∗≡dZv+λvdtanddZ∗d≡dZd+λddtare the new
Brownian motions under the risk-neutral probabilityQandCov(dZv∗,dZ∗d)=ρvddt.
Applying Ito’s lemma, we can reach the equation for the ratio-price simulationP=VD
under the risk-neutral measureQ:

dP

P

=(−δ+σd^2 −σvσdρvd)dt+σvdZ∗v−σddZ∗d. (7)

Applying the log-transformation forDT, under the probabilityQ, it results in:

DT=D 0 exp{(r−δd)T}·exp

(

−

σd^2

2

T+σdZd∗(T)

)

. (8)

We h ave t h at U ≡

(

−

σd^2

2 T+σdZ

∗

d(T)

)

∼ N

(

−

σd^2

2 T,σd

√

T

)

and

therefore exp(U)is a log-normal whose expectation value isEQ

[

exp(U)

]

=

exp

(

−

σ^2 d

2 T+

σd^2

2 T

)

=1. So, by Girsanov’s theorem, we can define the new prob-

ability measure

∼

Qequivalent toQand the Radon-Nikodym derivative is:

d

∼

Q

dQ

=exp

(

−

σd^2

2

T+σdZ∗d(T)

)

. (9)

Hence, using Equation (8), we can write:

DT=D 0 e(r−δd)T·

d

∼

Q

dQ

. (10)

By the Girsanov theorem, the processes:

dZˆd=dZ∗d−σddt, (11)

dZˆv=ρvddZˆd+

√

1 −ρ^2 vddZ′, (12)